The primary uses of Lemma REF  are with the case \(f \equiv 1\)  and setting \(f\)  as \(f_n\) . With only a little more work via the Borel-Cantelli lemma, [1]} shows \(\mathbb {P}( \limsup \mathcal {A}_n^c ) = 0\)  if \((k_n \log k_n)^2 / n \rightarrow 0\) . But for our purposes we want to keep track of the various rates of convergences and thus we do not invoke this result. Note that a look into the proof shows the result holds regardless of whether \(f\)  is dependent on \(n\) . Indeed, Lemma REF  motivates the definitions
\({\begin{array}{c}\hat{\Pi }_{k_n}:= 1_{\mathcal {A}_n} \frac{1}{2\pi \iota } \int _{\hat{\mathcal {C}}_n} (z - \Gamma _n)^{-1} \,\mathrm {d}z\equiv 1_{\mathcal {A}_n} \frac{1}{2\pi \iota } \int _{\mathcal {C}_n} (z - \Gamma _n)^{-1} \,\mathrm {d}z \\\Gamma _n^\dagger := 1_{\mathcal {A}_n} \frac{1}{2\pi \iota } \int _{\hat{\mathcal {C}}_n} f_n(z) (z - \Gamma _n)^{-1} \,\mathrm {d}z\equiv 1_{\mathcal {A}_n} \frac{1}{2\pi \iota } \int _{\mathcal {C}_n} f_n(z) (z - \Gamma _n)^{-1} \,\mathrm {d}z\end{array}}\) 
